({{, (RefLink[FunctionDomain, paclet:ref/FunctionDomain])[f,x] 
finds the largest domain of definition of the real function f of the variable x. }, {, (RefLink[FunctionDomain, paclet:ref/FunctionDomain])[f,x,dom]
considers f to be a function with arguments and values in the domain dom.}, {, (RefLink[FunctionDomain, paclet:ref/FunctionDomain])[funs,vars,dom]
finds the largest domain of definition of the mapping funs of the variables vars. }, {, (RefLink[FunctionDomain, paclet:ref/FunctionDomain])[{funs,cons},vars,dom]
finds the domain of funs with the values of vars restricted by constraints cons.}})

OpenersNotesSection(Spacer1)Details and Options

funs should be a list of functions of variables vars.

Possible values for dom are (RefLink[Reals, paclet:ref/Reals]) and (RefLink[Complexes, paclet:ref/Complexes]). The default is (RefLink[Reals, paclet:ref/Reals]).

If dom is (RefLink[Reals, paclet:ref/Reals]) then all variables, parameters, constants, and function values are restricted to be real.

cons can contain equations, inequalities, or logical combinations of these.

The following options can be given:

OpenersPrimaryExamplesSection(Spacer1)Examples  (8)

OpenersExampleSection0(Spacer1)Basic Examples  (2)

Find the largest domain of definition of a real function:

FunctionDomain[x/(x^4-1),x]

The largest domain of definition of a complex function:

FunctionDomain[Tanh[x],x,Complexes]

OpenersExampleSection1(Spacer1)Scope  (4)

Real univariate functions:

FunctionDomain[x/(x^2-1),x]

FunctionDomain[Log[Tan[x]-1],x]

FunctionDomain[Gamma[1-x^2],x]

Domain restricted by constraints:

FunctionDomain[{Log[Tan[x]-1],x^2<1},x]

Complex univariate functions:

FunctionDomain[x/(x^2+1),x,Complexes]

FunctionDomain[Log[Tan[x]-1],x,Complexes]

FunctionDomain[Log[Zeta[x]],x,Complexes]

Real multivariate functions:

FunctionDomain[(x+y)/(x^2-y^2),{x,y}]

FunctionDomain[Beta[a,Sqrt[x]],{a,x}]

Complex multivariate functions:

FunctionDomain[ArcTan[x,y]/Tanh[z],{x,y,z},Complexes]

FunctionDomain[BesselJ[x,1+Gamma[y]],{x,y},Complexes]

OpenersExampleSection2(Spacer1)Options  (2)

OpenersExampleSubsection0(Spacer1)GeneratedParameters  (1)

(RefLink[FunctionDomain, paclet:ref/FunctionDomain]) may introduce new parameters to represent the domain:

FunctionDomain[Log[Sin[x]],x]

Use (RefLink[GeneratedParameters, paclet:ref/GeneratedParameters]) to control how the parameters are generated:

FunctionDomain[Log[Sin[x]],x,GeneratedParameters->(Subscript[k,#]&)]

OpenersExampleSubsection1(Spacer1)Method  (1)

By default, domains of real univariate functions are given in a reduced form:

FunctionDomain[1/Log[Exp[1/x]-x^2],x]

Domains of other functions are not reduced:

FunctionDomain[Sqrt[x^2-x y+y^2],{x,y}]

Use (RefLink[Method, paclet:ref/Method]) to specify whether the domain should be given in a reduced form:

FunctionDomain[1/Log[Exp[1/x]-x^2],x,Method->{"Reduced"->False}]

FunctionDomain[Sqrt[x^2-x y+y^2],{x,y},Method->{"Reduced"->True}]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[FunctionRange, paclet:ref/FunctionRange])  ▪  (RefLink[Reduce, paclet:ref/Reduce])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Assumptions and Domains, paclet:guide/AssumptionsAndDomains])}, {▪(RefLinkPlain[Inequalities, paclet:guide/Inequalities])}, {▪(RefLinkPlain[Calculus, paclet:guide/Calculus])}})}})